In this paper, we investigate, in the framework of the topological approach to black hole thermodynamics, using the generalized off-shell Helmholtz free energy, the topological numbers of the static multi-charge AdS black holes in four- and five-dimensional gauged supergravities. We find that the topological number of the static-charged AdS black holes in four-dimensional Kaluza-Klein (K-K) gauged supergravity theory is W = 0, while that of the static-charged AdS black holes in four-dimensional gauged –iX ⁰ X ¹-supergravity and STU gauged supergravity theories, and five-dimensional Einstein-Maxwell-dilaton-axion (EMDA) gauged supergravity and STU gauged supergravity, and five-dimensional static-charged AdS Horowitz-Sen black hole are both W = 1. Furthermore, we observe a novel temperature-dependent thermodynamic topological phase transition that can happen in the four-dimensional static-charged AdS black hole in EMDA gauged supergravity theory, the four-dimensional static-charged AdS Horowitz-Sen black hole, and the five-dimensional static-charged AdS black hole in K-K gauged supergravity theory. We believe that the novel temperature-dependent thermodynamic topological phase transition could help us better understand black hole thermodynamics and, further, shed new light on the fundamental nature of gauged supergravity theories.

Via the semiclassical approximation method, we study the $1/2$-spin fermion tunneling from a higher-dimensional black hole. In our work, the Dirac equations are transformed into a simple form, and then we simplify the fermion tunneling research to the study of the Hamilton-Jacobi equation in curved space-time. Finally, we get the fermion tunneling rates and the Hawking temperatures at the event horizon of higher-dimensional black holes. We study fermion tunneling of a higher-dimensional Schwarzschild black hole and a higher-dimensional spherically symmetric quintessence black hole. In fact, this method is also applicable to the study of fermion tunneling from four-dimensional or lower-dimensional black holes, and we will take the rainbow-Finsler black hole as an example in order to make the fact explicit.

The tunneling behavior of fermions with half-integral spin from a higher dimensional charged anti-de Sitter (AdS) black hole in de Rham, Gabadadze and Toll

In this paper, taking into account a black hole solution of Finsler theory, we study the remnant and its phase transition close to the Planck scale. Based